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Answer :
To factor the expression [tex]\(x^4 - 4x^2 - 45\)[/tex] completely, we can use substitution and polynomial factoring techniques:
1. Substitution:
Let's substitute [tex]\(y = x^2\)[/tex]. This transforms the expression [tex]\(x^4 - 4x^2 - 45\)[/tex] into a quadratic in terms of [tex]\(y\)[/tex]:
[tex]\[
y^2 - 4y - 45
\][/tex]
2. Factoring the quadratic:
We need to factor [tex]\(y^2 - 4y - 45\)[/tex]. To do this, look for two numbers that multiply to [tex]\(-45\)[/tex] (the constant term) and add to [tex]\(-4\)[/tex] (the coefficient of [tex]\(y\)[/tex]).
These numbers are [tex]\(-9\)[/tex] and [tex]\(5\)[/tex], because:
- [tex]\((-9) \times 5 = -45\)[/tex]
- [tex]\((-9) + 5 = -4\)[/tex]
So, we can factor the quadratic as:
[tex]\[
(y - 9)(y + 5)
\][/tex]
3. Substitute back to [tex]\(x\)[/tex]:
Recall that [tex]\(y = x^2\)[/tex]. Substitute back to get expressions in terms of [tex]\(x\)[/tex]:
[tex]\[
(x^2 - 9)(x^2 + 5)
\][/tex]
4. Further factor [tex]\(x^2 - 9\)[/tex]:
Notice that [tex]\(x^2 - 9\)[/tex] is a difference of squares:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
5. Final Factored Form:
So, the complete factorization of the original expression is:
[tex]\[
(x - 3)(x + 3)(x^2 + 5)
\][/tex]
And that's the complete factorization of the expression [tex]\(x^4 - 4x^2 - 45\)[/tex].
1. Substitution:
Let's substitute [tex]\(y = x^2\)[/tex]. This transforms the expression [tex]\(x^4 - 4x^2 - 45\)[/tex] into a quadratic in terms of [tex]\(y\)[/tex]:
[tex]\[
y^2 - 4y - 45
\][/tex]
2. Factoring the quadratic:
We need to factor [tex]\(y^2 - 4y - 45\)[/tex]. To do this, look for two numbers that multiply to [tex]\(-45\)[/tex] (the constant term) and add to [tex]\(-4\)[/tex] (the coefficient of [tex]\(y\)[/tex]).
These numbers are [tex]\(-9\)[/tex] and [tex]\(5\)[/tex], because:
- [tex]\((-9) \times 5 = -45\)[/tex]
- [tex]\((-9) + 5 = -4\)[/tex]
So, we can factor the quadratic as:
[tex]\[
(y - 9)(y + 5)
\][/tex]
3. Substitute back to [tex]\(x\)[/tex]:
Recall that [tex]\(y = x^2\)[/tex]. Substitute back to get expressions in terms of [tex]\(x\)[/tex]:
[tex]\[
(x^2 - 9)(x^2 + 5)
\][/tex]
4. Further factor [tex]\(x^2 - 9\)[/tex]:
Notice that [tex]\(x^2 - 9\)[/tex] is a difference of squares:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
5. Final Factored Form:
So, the complete factorization of the original expression is:
[tex]\[
(x - 3)(x + 3)(x^2 + 5)
\][/tex]
And that's the complete factorization of the expression [tex]\(x^4 - 4x^2 - 45\)[/tex].
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